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# Full text of "Analytical Mechanics"

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```WORK                              '          177
The modulus of elasticity X is not a definite constant f 01 a given gas, because the value of -~ depends upon the temperature and the amount of heat of the gas. Therefore the
fi T)
state of a gas for which -~- is calculated should be stated in
dv
order that the value of X may have any meaning at all There are two states for which X is calculated, namely, the isothermal and the adiabatic states.
145. Isothermal Elasticity. — When the compression is isothermal
pv = k
and                              ? = -A
Therefore the isothermal elasticity of a gas numerically equals the pressure.
146.   Adiabatic Elasticity. — When the gas is compressec adiabatically
pv^ = k
and                           ^ = - kyv-^-i
dv
= - yptr1. •'-. x== TP-                                      ®.
147.   Torsional Rigidity of a Shaft. — Suppose the uppe] end of the cylinder of Fig. 97 to be rotated about the axij of the cylinder through an angle 0, while the lower end if fixed, and consider the stresses and the strains in the cylin der.    It is evident that the strain is nil at the axis and in creases uniformly with the distance from the axis.    Furthe] the strain is nil at the lower base and increases uniformly with the distance from it.    Since Hooke's law holds thes< statements are true with regard to the stress in the cylinder```